Interactions between Moduli Spaces, Non-Commutative Algebra, and Deformation Theory.

数学领域的许多伟大成就都源于看似脱节的研究领域之间的联系，使一个领域的技术和见解能够为另一个领域的问题带来新的曙光。其中一个例子就是使用非交换代数来学习几何。将代数和几何洞察力结合起来，通常可以将结果扩展到更自然的泛化水平，突破几何设置的限制，并从几何中产生有趣的代数结构。这种方法在研究奇点的分解方面特别成功。给出了具有非交换解释的重构代数的有理曲面奇点的极小解的一个例子。这些有理表面奇点的最小分辨率所具有的另一个特征是，它们有一个特别迷人和美丽的几何变形理论，然而目前这还不能从非交换的观点来理解。重构代数的变形理论被期望与几何情况有内在的联系，因此应该反映其有趣的特征，同时从非交换的观点提供新的见解。本研究试图通过在几何变形理论和非交换变形理论之间建立一座桥梁来理解这样的例子。这将涉及发展构造非交换代数变形的技术，以及产生从非交换代数作为模空间恢复几何变形的方法。它还将涵盖一般情况，例如移动到光滑品种的设置之外，这将在最小模型程序中的3倍结构等领域产生广泛的新应用。

Many great successes within mathematics arise from linking between seemingly disjoint fields of research, allowing techniques and insights developed in one area to shine a new light on problems in another. One example of this is the use of non-commutative algebra to study geometry. Combining both algebraic and geometric insight often allows results to be extended to more natural levels of generalisation, breaking out of restrictions imposed by geometric settings and producing interesting algebraic structures from the geometry. This approach has been particularly successful in the study of resolutions of singularities.An example is provided by minimal resolutions of rational surface singularities having a non-commutative interpretation as reconstruction algebras. Another feature that these minimal resolutions of rational surface singularities possess is that they have a particularly fascinating and beautiful geometric deformation theory, however currently this is not understood from a non-commutative viewpoint. The deformation theory of the reconstruction algebras is expected to be intrinsically linked to the geometric case and so should mirror its interesting features while offering new insights from a non-commutative viewpoint.This research seeks to understand examples such as this by building a bridge between the geometric and non-commutative deformation theory. This will involve developing techniques to construct deformations of non-commutative algebras and producing methods of recovering geometric deformations from non-commutative ones as moduli spaces. It will also encompass general situations, such as moving outside the setting of smooth varieties, which will generate a wide range of new applications in areas such as the construction of 3-folds in the minimal model program.